Link between bipartite and general unicellular toroidal maps via slit--slide--sew bijections

Abstract

We relate general maps to bipartite maps through a bijection of type slit-slide-sew. We provide an involution on arbitrary genus maps with even degree faces. This allow a full interpretation of the relation between general and bipartite maps, in the case of genus $1$ maps with a unique face. The main tool is the use of rotations along well-chosen specific loops. Once a noncontractible simple loop is given, one slit along it, slide one notch, and sew back. This mildly modifies the structure of the map along the loop, changing the parity of the length of other loops crossing it. In the case of unicellular toroidal maps, the simple structure of noncontractible loops makes it possible to fully relate general maps to bipartite maps.

Figures (8)

• Figure 1: Map of genus $1$ with even degree faces. The root is the corner represented by the red arrow.
• Figure 2: Main involution. The origin of the loops $\boldsymbol{\lambda}$ and $\tilde{\boldsymbol{\lambda}}$ are indicated with purple vertices. When $\varepsilon=+$, we "screw" the map one notch along the distinguished loop. We "unscrew" one notch along the loop if $\varepsilon=-$.
• Figure 3: Change of parity in the length of a path crossing the sliding loop.
• Figure 4: Rightmost noncontractible simple loop of the map from Figure \ref{['map']}. The loop $\boldsymbol{\lambda}_\mathbf{m}$ is the loop part (in purple) of the rightmost path (in thicker lines) from the root consisting of a simple path concatenated with a noncontractible simple loop.
• Figure 5: Preservation of the rightmost loop by the involution $\Phi$. We have, in $\mathbf{m}$ on the left, the paths $\mathbf{p}$ and $\boldsymbol{\lambda}_\mathbf{m}$, which are mapped through $\Phi$, in $\tilde{\mathbf{m}}$ on the right, respectively into $\tilde{\mathbf{p}}$ and $\tilde{\boldsymbol{\lambda}}$. The rightmost path $\tilde{\boldsymbol{\wp}}$ of $\tilde{\mathcal{S}}$ in $\tilde{\mathbf{m}}$ is in red: after leaving $\tilde{\mathbf{p}}\bullet\tilde{\boldsymbol{\lambda}}$, it might either not intersect $\tilde{\boldsymbol{\lambda}}$ (top line), or reach $\tilde{\boldsymbol{\lambda}}$ from its right (middle line), or reach $\tilde{\boldsymbol{\lambda}}$ from its left (bottom line). In each case, tracing it back in $\mathbf{m}$ provides a path, in green, belonging to $\mathcal{S}$ and more to the right than $\mathbf{p}\bullet\boldsymbol{\lambda}_\mathbf{m}$.

Theorems & Definitions (12)

• Proposition 1
• Proposition 2
• proof
• Remark 1
• Theorem 3
• proof
• Theorem 4
• proof : Proof of Theorem \ref{['thmspec']}, Bijections \ref{['speca']} to \ref{['specf']}
• proof : Proof of Theorem \ref{['thmspec']}, Bijection \ref{['specg']}
• Corollary 5